Fix an algebraically closed field $\mathbb{ F}$ and an integer $d \geq 3$. Let $\text{ Mat}_{ d+1}(\mathbb{ F})$ denote the $\mathbb{ F}$-algebra consisting of the $(d+1) \times (d+1)$ matrices that have all entries in $\mathbb{ F}$. We consider a pair of diagonalizable matrices $A,A^*$ in $\text{ Mat}_{ d+1}(\mathbb{ F})$, each acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{ Mat}_{ d+1}(\mathbb{ F})$. For a Leonard pair $A,A^*$ there is a nonzero scalar $q$ that is used to describe the eigenvalues of $A$ and $A^*$. In the present paper we find all Leonard pairs $A,A^*$ in $\text{ Mat}_{ d+1}(\mathbb{ F})$ such that $A$ is lower bidiagonal with subdiagonal entries all $1$ and $A^*$ is irreducible tridiagonal, under the assumption that $q$ is not a root of unity. This gives a partial solution of a problem given by Paul Terwilliger.
Categories: math.RA